The radius and the height of a right circular cone are each increased by 20%. By what percentage does the volume of the cone increase?

Introduction / Context:
This quantitative aptitude question examines how changes in linear dimensions affect the volume of a three dimensional solid, specifically a right circular cone. It highlights that when more than one dimension changes, the resulting percentage change in volume is not simply the sum of the individual percentage changes. Instead, the combined effect must be calculated using the volume formula.

Given Data / Assumptions:

• We have a right circular cone with initial radius r and height h.
• The radius is increased by 20%, so the new radius r_new = 1.2 * r.
• The height is also increased by 20%, so the new height h_new = 1.2 * h.
• The original volume and new volume are to be compared.
• We must find the percentage increase in volume.

Concept / Approach:
The volume of a right circular cone is:
V = (1 / 3) * pi * r^2 * h.
The new volume after changes in radius and height is:
V_new = (1 / 3) * pi * (r_new)^2 * h_new.
Substituting r_new = 1.2r and h_new = 1.2h, we can express V_new as a multiple of V. The factor multiplying V gives the overall volume scale change, and from this we can compute the percentage increase.

Step-by-Step Solution:
Step 1: Write the original volume: V = (1 / 3) * pi * r^2 * h. Step 2: The new radius is r_new = 1.2r. Step 3: The new height is h_new = 1.2h. Step 4: Compute the new volume: V_new = (1 / 3) * pi * (1.2r)^2 * (1.2h). Step 5: Simplify (1.2r)^2 = 1.2^2 * r^2 = 1.44 * r^2. Step 6: So V_new = (1 / 3) * pi * 1.44 * r^2 * 1.2 * h. Step 7: Combine the constants: 1.44 * 1.2 = 1.728. Step 8: Hence V_new = 1.728 * (1 / 3) * pi * r^2 * h = 1.728 * V. Step 9: This means the new volume is 1.728 times the original volume. Step 10: The increase factor above 1 is 1.728 − 1 = 0.728. Step 11: Convert this factor to a percentage: 0.728 * 100% = 72.8%.

Verification / Alternative check:
We can use an approximate method to cross check. A 20% increase in radius alone would increase the term r^2 by about 44% (since 1.2^2 = 1.44), and an additional 20% increase in height would further scale volume by 1.2. Combining these effects: 1.44 * 1.2 = 1.728, which indeed represents a 72.8% increase over the original. This matches the detailed calculation.

Why Other Options Are Wrong:
20% and 20.5%: These mistakenly treat the volume change as if only one dimension changed or ignore the squared dependence on radius.
62%: This may come from adding 44% (due to r^2) and 18% approximately, but it is still incorrect because the effects are multiplicative, not additive.
Any other values different from 72.8% do not align with V_new = 1.728 * V computed using the exact formula.

Common Pitfalls:
A common error is to assume that if both radius and height increase by 20%, the volume also increases by 40%. This ignores the fact that volume depends on r^2 and h, so the changes combine multiplicatively. Students also sometimes miscalculate 1.2^2 and obtain incorrect intermediate values. Carefully writing out the algebra and doing the multiplication in steps helps avoid such mistakes.

Final Answer:
The volume of the cone increases by 72.8%.